
where \(w_k = (\xi _{k+1} - \xi _{k-1})/2 \ (\mathrm {mod}\ 1)\)  are the quadrature nodes for the grid.
We can now use (REF ) to approximate \(\hat{v}\)  and \(\hat{v}\)  and thereby approximate the solution (REF ).
The sums (REF ) can be computed efficiently with
the non-uniform discrete Fourier transform (NUDFT–II, [1]}, [2]}).
